Linear and nonlinear equations for beams and degenerate plates with multiple intermediate piers

TL;DR

This paper develops a comprehensive theory for hinged beams and degenerate plates with multiple piers, analyzing linear and nonlinear stability, eigenvalues, and optimal pier placement for structural stability, with applications to suspension bridges.

Contribution

It introduces a full variational and spectral analysis for structures with piers, including nonlinear instability and optimal pier positioning, extending existing models to more complex configurations.

Findings

Eigenvalues depend on pier positions and influence oscillation modes.

Optimal pier placement enhances structural stability against nonlinear instabilities.

Torsional instability analyzed with Floquet theory and numerical simulations.

Abstract

A full theory for hinged beams and degenerate plates with multiple intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem in one dimension. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is underlined. A complete spectral theorem is then proved, explicitly determining the eigenvalues on varying of the position of the piers and exhibiting the fundamental modes of oscillation. The obtained eigenfunctions are used to tackle the study of the nonlinear evolution problem in presence of different nonlinearities, focusing on the appearance of linear and (a suitable notion of) nonlinear instability, with a twofold goal: finding the most reliable nonlinearity describing the oscillations of real structures and determining the position of the piers that maximizes the stability of the structure. The last part of the paper is devoted to the study of degenerate plate models, where the structure is composed by a central beam moving vertically and by a continuum of cross-sections free to rotate around the beam and allowed to deflect from the horizontal equilibrium position. The torsional instability of the structure is investigated, taking into account the impact of different nonlinear terms aiming at modeling the action of cables and hangers in a suspension bridge. Again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out both by means of analytical tools, such as Floquet theory, and numerical experiments. Several open problems and possible future developments are presented.